This dissertation deals with multidimensional versions of the classical Khintchine conjecture on almost-everywhere convergence of certain averages originating in the theory of uniform distribution. We study analogous actions of multiplicative semigroups of rings and investigate both negative results extending Marstrand's answer to Khintchine's conjecture, and positive results answering related questions. We first show that Lp-convergence holds very broadly for almost any sequence of averages for these actions. Next, we show that almost-everywhere convergence of averages taken over additive Følner sequences fails. Finally, we investigate what kinds of averages for these actions give pointwise convergence almost everywhere.